Invitations Spring 2024: lectures in Copenhagen and Lund Universities



Solving the Invariant Subspace Problem

PER ENFLO  On the invariant subspace problem in Hilbert spaces  youtube >> 12:00 - 13:00

Craiova, Romania, the 20 th October 2023

"Invited to participate as a soloist in the Concert no. 1 in C major for piano and orchestra, op. 15 of Ludwig van Beethvoven was Per Enflo.


A Swede settled in the United States, this respectable gentleman is not only a pianist so good that he has touched all the meridians of the world with his art, but also a renowned mathematician.


Yes, mathematician.


Two fields that seem so far apart, but which, at the highest level, approach asymptotically. Let us think only of Pythagoras and the sacred importance he attached to music.



Per Enflo has not only a remarkable destiny, but also a long-lasting one, which I wish twice as long.


At 80, he still has the energy to let himself be swept up in the gaiety of Beethoven's composition with the enthusiasm of a toddler (until his piece began, he seemed to be struggling to resist the temptation to touch the keyboard).


What is the secret of such a zest for life at such an advanced age?


But perhaps that is not the question, but the answer.


With his lordship's permission, I will also draw a parallel between the way he crossed his hands on the piano keys and the way of multiplying the means and extremes of a proportion.

Photos by my friend Daniel Botea, who, although he is incredibly old, enjoyed the same as me, like a child that is, all the wonders that the concert offered.

Autor: Iris Gottlieb (sursa – www.vox.com)

Google translation from original Romanian, Facebook:


IWOTA 2023 July 31 - August 4, 2023, 34th International Workshop on Operator Theory and its Applications, University of Helsinki, Finland read more >>

  • Semi-plenary session Tuesday August 1st at 12:15 – 1:00  On the Invariant Subspace Problem in Hilbert space.
  • Concert Thursday August 3rd at 6:45  Beethoven, Schubert and Schumann at the Conference Dinner

On the invariant subspace problem in Hilbert spaces

Per H. Enflo     Available online >> arXiv:2305.15442 [pdf,ps,other]

In this paper we show that every bounded linear operator T on a Hilbert space H has a closed non-trivial invariant subspace.  Submitted 24 May, 2023;originally announced May 2023.  Comments:13 pages MSC Class:47A15; 47B02


A method to construct invariant subspaces for a general operator T on Hilbert space is presented. It represents a new direction of my "method of extremal vectors", first presented in 1998 in Ansari-Enflo [1].

The Main Construction of the method gives a non-cyclic vector of T by gradual approximation by "almost non-cyclic" vectors. There are reasons, why the Main Construction cannot work for some weighted shifts. But when the Main Construction fails, one gets, by using the information obtained, invariant subspaces of T, similar to those of weighted shifts.

The sequence y(n) of "almost non-cyclic" vectors follows the formula y(n+1) = y(n) + r(T) y(n), which will allow for efficient use of elementary Fourier Analysis. The more general formula y(n+1) = y(n) + z would lead to difficult problems concerning the relation between T and T*, problems which may be of interest in themselves.


  1. S.Ansari, P.Enflo, "Extremal vectors and invariant subspaces", Transactions of Am. Math. Soc. Vol. 350 no.2, 1998, pp.539-558

Invariant Subspace Problem in Hilbert Spaces:

Exploring Applications in Quantum Mechanics, Control Theory, Operator Algebras, Functional Analysis and Accelerator Physics - Mostafa Behtouei arXiv:2306.17023

"Per H. Enflo’s recent claim of solving the invariant subspace problem in his recent publication titled ”On the invariant subspace problem in Hilbert spaces.” [4] has generated excitement and anticipation in the mathematical community, as his work could represent a major breakthrough in the field. Enflo’s solution, if validated,would be a groundbreaking achievement in mathematics, resolving a longstanding problem and advancing our understanding of bounded linear operators on Hilbert spaces. Its impact would be significant and far-reaching." From: Mostafa Behtouei, 29 Jun 2023

Interdisciplinary Mathematics Institute (IMI)

25 de mayo de 2023.Boletín del IMI.

A potential definitive answer to the invariant subspace problem in Hilbert spaces. Prof. Per H. Enflo (Kent State University, USA) has just announced via his collaborators from the Interdisciplinary Mathematics Institute (IMI) that he might have just potentially solved the long standing Invariant Subspace Problem in Hilbert Spaces

A history of solving some famous problems in mathematical analysis

This paper provides a historical view of what functional analysis is, a biography of Per H. Enflo and a short view of four fundamental problems in the geometry of Banach spaces.

It gives the story of how Enflo, after several years of work, managed to solve three of these problems. It is a story of trying many ideas, testing them on simpler problems, trying to combine different techniques for handling different aspects of the problems, making many failed attempts and finally succeeding.

In 1966 Enflo found a general strategy for solving many problems in infinite dimensions, by combining appropriate finite-dimensional techniques with induction procedures. When applied to the old, fundamental problems in functional analysis, this has led to new concepts, problems, techniques and results in analysis. These developments have been applied also in several other areas of mathematics and computer science.

Department of Mathematical Sciences, Kent State University, Kent, OH, USA
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran
Instituto de Matemática Interdisciplinar (IMI), Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid, Spain

When did you make the decision to go into math instead of music?

I have never made, and will never make, any such decision

  • "Mathematics and Music - What is the connection?"
    Lecture >> 31 August 2021, Science Festival Strömstad Academy Slides >
  • Beethoven last piano sonata Concert >> Strömstad Academy's Scientific Festival 2021

  • Mozart - Beethoven - Chopin Concert >>  Jubilee Congress of Polish Mathematicians on the 100th Anniversary of Polish Mathematical Society (2019, Krakow)

  • "An infinite-dimensional version of Hilbert's fifth problem with connections to non-linear geometry of Banach spaces"

    Lecture December 2021  Slides >> to Holiday Lecture by Per Enflo  via Zoom - Jani Virtanen (Reading and Helsinki) & Håkan Hedenmalm (KTH and Reading) 

Mathematics >>     Music >>     Interviews & Reviews >>

Per Enflo - mathematician and pianist - was the last laureate of a prize

for solving a scientific problem from the Scottish Book.

Contact  +46(0)721-808 870       per.enflo at gmail.com       www.perenflo.com © 2011